cp's OEIS Frontend

A023543 Convolution of natural numbers with A023533.

%I A023543 #11 Jul 16 2022 03:53:34
%S A023543 1,2,3,5,7,9,11,13,15,18,21,24,27,30,33,36,39,42,45,49,53,57,61,65,69,
%T A023543 73,77,81,85,89,93,97,101,105,110,115,120,125,130,135,140,145,150,155,
%U A023543 160,165,170,175,180,185,190,195,200,205,210,216,222,228
%N A023543 Convolution of natural numbers with A023533.
%H A023543 G. C. Greubel, <a href="/A023543/b023543.txt">Table of n, a(n) for n = 1..5000</a>
%F A023543 From _G. C. Greubel_, Jul 15 2022: (Start)
%F A023543 a(n) = Sum_{j=1..floor((n+1)/2)} (n - j + 1)*A023533(j).
%F A023543 a(n) = (m+2)*(n+1) - binomial(n+4, 4), for binomial(n+3, 3) - 2 <= m <= binomial(n+4, 3) - 3, and n >= 1, with a(1) = 1, a(2) = 2. (End)
%t A023543 Join[{1,2}, Table[(m+2)*(n+1) -Binomial[n+4,4], {n,6}, {m, Binomial[n+3,3] -2, Binomial[n+4,3] -3}]]//Flatten (* _G. C. Greubel_, Jul 15 2022 *)
%o A023543 (Magma)
%o A023543 A023533:= func< n | Binomial(Floor((6*n-1)^(1/3)) +2, 3) ne n select 0 else 1 >;
%o A023543 [(&+[A023533(k)*(n+1-k): k in [1..Floor((n+1)/2)]]): n in [1..100]]; // _G. C. Greubel_, Jul 15 2022
%o A023543 (SageMath)
%o A023543 [1,2]+flatten([[(m+2)*(n+1) - binomial(n+4,4) for m in (binomial(n+3,3)-2 .. binomial(n+4,3)-3)] for n in (1..6)]) # _G. C. Greubel_, Jul 15 2022
%Y A023543 Cf. A000027, A023533.
%K A023543 nonn
%O A023543 1,2
%A A023543 _Clark Kimberling_
%E A023543 Title updated by _Sean A. Irvine_, Jun 06 2019